The issue (after hours of looking) is that OmniSharp requires the Mono runtime. I had pointed at the coreclr (1.0.0-rc1-update1, coreclr x64, darwin) the other day because Mono wasn't compatible with SSL-to-Azure connections. To keep things clean, I had just uninstalled the Mono runtime, hoping to just use the coreclr.

The process for this: I first found this Git post: No intellisense on Linux for ASP .NET 5 coreclr x64 1.0.0.-rc1-final. I followed the instructions and showed the OmniSharp output (how to: Command, Shift, p in Visual Studio Code to open prompt, then type output, select Toggle Output and then change the view to OmniSharp using the dropdown). This output, when trying intellisense, showed a lot of errors being generated.

I then searched those errors and found this question/answer Visual Studio Code Designhostmanager Failed. Apparently, I needed to download the Mono runtime, specifically (in my case), Beta5, per the comment from Matt. To do this I ran dnvm install 1.0.0-beta5 in my terminal on OSX and the runtime installed and set Mono as the currently used runtime. Then, I ran dnu restore to make sure all of the proper library references were cleaned up. Finally, Visual Code prompted me for an additional restore, which I ran as well. Here is my final output (please ignore the extra Mono runtimes. I was playing around and wasn't exactly sure what the specific differences were between them):